Espaces BMO, inégalités de Paley et multiplicateurs idempotents

• Autorzy: Hubert Lelièvre
• Języki publikacji: FR EN

Abstrakty

EN

Generalizing the classical BMO spaces defined on the unit circle 𝕋 with vector or scalar values, we define the spaces $BMO_{ψ_{q}}(𝕋)$ and $BMO_{ψ_{q}}(𝕋,B)$, where $ψ_{q}(x) = e^{x^q} -1$ for x ≥ 0 and q ∈ [1,∞[, and where B is a Banach space. Note that $BMO_{ψ_{1}}(𝕋) = BMO(𝕋)$ and $BMO_{ψ_{1}}(𝕋,B) = BMO(𝕋,B)$ by the John-Nirenberg theorem. Firstly, we study a generalization of the classical Paley inequality and improve the Blasco-Pełczyński theorem in the vector case. Secondly, we compute the idempotent multipliers of $BMO_{ψ_{q}}(𝕋)$. Pisier conjectured that the supports of idempotent multipliers of $L^{ψ_{q}}(𝕋)$ form a Boolean algebra generated by the periodic parts and the finite parts for 2 < q < ∞. We confirm this conjecture with $L^{ψ_{q}}(𝕋)$ replaced by $BMO_{ψ_{q}}(𝕋)$.

Kategorie tematyczne

• 42A55: Lacunary series of trigonometric and other functions; Riesz products
• 42A45: Multipliers
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46B20: Geometry and structure of normed linear spaces
• 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 123
• Numer: 3
• Strony: 249-274

Daty

wydano

1997

otrzymano

1996-02-14

poprawiono

1996-09-27

Twórcy

autor

Hubert Lelièvre

• Equipe d'analyse, URA 754, Université Pierre et Marie Curie-Paris 6, Tour 46-0, Boîte 186, 4, place Jussieu 75252 Paris Cedex 05, France

Bibliografia

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• [CW] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
• [GM] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York, 1979.
• [H] H. Helson, Note on harmonic functions, Proc. Amer. Math. Soc. 4 (1953), 686-691.
• [Kl] I. Klemes, Idempotent multipliers of $H^1(T)$, Canad. J. Math. 39 (1987), 1223-1234.
• [L] H. Lelièvre, Espaces BMO et multiplicateurs idempotents, thèse de doctorat de l'Université Paris 6, 1995.
• [Pe] A. Pełczyński, Commensurate sequences of characters, Proc. Amer. Math. Soc. 104 (1988), 525-531.
• [Pi1] G. Pisier, Les inégalités de Khintchine-Kahane d'après C. Borel, Séminaire sur la géométrie des espaces de Banach 1977-1978, exposé VII, Ecole Polytechnique, Centre de Mathematiques, 1978.
• [Pi2] G. Pisier, De nouvelles caractérisations des ensembles de Sidon, in: Mathematical Analysis and Applications, Part B, Adv. in Math. Suppl. Stud. 7B, Academic Press, 1981, 685-726.
• [Pi3] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
• [Pi4] G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis 1985, Lecture Notes in Math. 1206, Springer, 1996, 167-241.